Question

复数z=i+i²+i³+i⁴+…+i²⁰⁰⁷+i²⁰⁰⁸+i²⁰⁰⁹的值是______．

Answer 1

根据虚数单位i的性质：当n∈N时，i⁴ⁿ=1，i⁴ⁿ⁺¹=i，i⁴ⁿ⁺²=-1，i⁴ⁿ⁺³=-i，
z=（i+i²+i³+i⁴）+…+（i²⁰⁰⁵+i²⁰⁰⁶+i²⁰⁰⁷+i²⁰⁰⁸）+i²⁰⁰⁹
=0+…0+i
=i
故答案为：i